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show a graph where lim(x->-4-) f(x) is infinite and lim(x->-4+) is finite show a graph that has shows both of these

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Sure!

Step 1: Conceptual Understanding

The first thing to know here is that you're being asked to construct a graph of a function where the limit as x approaches -4 from the left (x->-4-) is negative infinity and the limit as x approaches -4 from the right (x->-4+) is four.

Step 2: Function Selection

The simple function that meets these conditions is y = 1/(x+4). This function has an infinite discontinuity at x = -4, creating an asymptote at that value. When x approaches -4 from the right (x->-4+), the denominator gets closer to zero from the positive side and y goes towards positive infinity (which is not what we need). So consider the negative of this function, y = -1/(x+4). Now, as x approaches -4 from the right (x->-4+), y goes towards zero which is exactly what we need. And as x approaches -4 from the left (x->-4-), y goes towards negative infinity.

Step 3: Drawing the Graph

Here are the steps to draw the graph of y = -1/(x+4):

1. Draw Cartesian axes, label the x-axis as x and the y-axis as f(x). Mark the point (-4,0) on x-axis.

2. To the right of -4, the graph of -1/(x+4) is a hyperbola opening downwards and approaching the x-axis (y = 0) but never quite reaching it hence creating a horizontal asymptote.

3. To the left of -4, the graph of -1/(x+4) starts from negative infinity at x=-4 and rises but never crosses the x-axis hence creating a vertical asymptote.

4. Signify the asymptote at x = -4 with a red dotted vertical line labeled 'x=-4'.

5. You can select random points on either side of x = - 4 in case you would like to show specific values on y.

There you have it! This graph should clearly illustrate the required limit behaviors of the function. When x approaches -4 from the left or right, the y values either decrease without bound or approach zero respectively.

User Hwen
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